OFFPRINT Transport in networks with multiple sources and sinks

We investigate the electrical current and flow (number of parallel paths) between two sets of n sources and n sinks in complex networks. We derive analytical formulas for the average current and flow as a function of n. We show that for small n, increasing n improves the total transport in the network, while for large n bottlenecks begin to form. For the case of flow, this leads to an optimal n∗ above which the transport is less efficient. For current, the typical decrease in the length of the connecting paths for large n compensates for the effect of the bottlenecks. We also derive an expression for the average flow as a function of n under the common limitation that transport takes place between specific pairs of sources and sinks. Copyright c © EPLA, 2008 Transport processes, such as electrical current, diffusion, and flow, are fundamental in physics, chemistry, and biology. Transport properties depend critically on the structure of the medium, and have been studied for a large variety of geometries [1]. Of current interest are situations where the transport occurs on a network. For example, information is transferred over computer or social networks, vehicles traverse transportation networks, and electrical current flows in power-grid networks. Therefore, understanding of mechanisms to increase transport effectiveness is of great importance. Transport properties usually have been investigated in the context of transport between a single pair comprising one source and one sink [2–8]. The quality of transport strongly depends on the degree (number of connections) of the source and the sink, whereas the rest of the network serves as an approximately resistance-free substrate for the transport process. Consequently, the transport was found to strongly depend on the degree distribution. In more realistic situations, transport takes place between many nodes simultaneously. For example, in peer-to-peer and other computer networks users exchange files in parallel over the network links. In transportation networks, vehicles travel between many sources and many destinations through the network infrastructure. The presence of many parallel transport processes on the same underlying network leads to interactions between (a)E-mail: [email protected] the different deliveries and a change in network efficiency. In this article we will quantify this phenomenon analytically and numerically, and show how different network usage leads to different behaviors. We reported some preliminary results in [9]. We focus on the class of non-directed, non-weighted model networks and we also study a real network. The first model is the Erdős-Rényi (ER) network, in which each link exists with independent small probability p. This leads to a Poisson degree distribution [10,11]. The second model is the scale-free (SF) network, characterized by a broad, power law degree distribution and was recently found to describe many natural systems [12–14]. The ensemble of SF networks we treat is the “configuration model”, in which node degrees are drawn from a power law distribution (see below), and then open links are connected [15]. We also compute the distribution of flows in a real network, the internet [16]. We consider a transport process between two randomly chosen, non-overlapping sets (sources and sinks) of nodes of size n each, where 1 n N/2, N is the total number of nodes. We focus on three explicit forms of transport, described below. i) Maximum flow (henceforth denoted flow) between the sources and sinks, when each link has unit capacity [8,9,16–19]. For non-weighted networks the flow is equivalent to the total number of disjoint paths (i.e. paths that do not share any edge) that connect the sources and sinks. Therefore, it quantifies any flow which does not